Limit laws of random simplex tree-child networks

2026/07/16 by Víctor J. Maciá, Benedikt Stufler
#math.CO #math.PR

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Abstract

We prove that the longer and shorter Sackin indices of a uniformly random simplex tree-child network with n taxa admit joint distributional limits after rescaling by n^-7/4. The limiting distributions are described by functionals of a Brownian excursion. We also identify the limiting law of the height after rescaling by n^-3/4, thereby answering a question of Zhang~(2022). Moreover, we establish sharp tail bounds for the height, which imply convergence of all moments in the above distributional limits. We further obtain a scaling limit for the entire height profile of the leaves. Finally, we determine the local limits of large simplex networks around the fixed root, a uniformly random vertex, and a uniformly random leaf.

Related

• The Sackin Index of Simplex Networks
• Limit Theorems for Patterns in Ranked Tree-Child Networks
• Local limits of large Galton-Watson trees rerooted at a random vertex
• Scaling limit of critical random trees in random environment
• Limits of random tree-like discrete structures